Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction
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1 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Javier Esparza 1 Jan Křetínský 2 Salomon Sickert 1 1 Fakultät für Informatik, Technische Universität München, Germany 2 IST Austria May 11, 2015 Salomon Sickert (TU München) 1 / 16
2 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Salomon Sickert (TU München) 2 / 16
3 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Salomon Sickert (TU München) 3 / 16
4 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction System with stochasticity and non-determinism expressed as a Markov decision process M Linear time property expressed as an LTL formula ϕ Non-deterministic Büchi automaton B Product M R to be analysed Deterministic Rabin automaton R Salomon Sickert (TU München) 3 / 16
5 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction System with stochasticity and non-determinism expressed as a Markov decision process M Linear time property expressed as an LTL formula ϕ Non-deterministic Büchi automaton B Safra Product M R to be analysed Deterministic Rabin automaton R Salomon Sickert (TU München) 3 / 16
6 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction System with stochasticity and non-determinism expressed as a Markov decision process M Linear time property expressed as an LTL formula ϕ Product M R to be analysed Deterministic (transition-based) Generalised Rabin automaton R Salomon Sickert (TU München) 4 / 16
7 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Salomon Sickert (TU München) 5 / 16
8 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Salomon Sickert (TU München) 5 / 16
9 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Product of several automata Salomon Sickert (TU München) 5 / 16
10 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Product of several automata Logical structure of the input formula is preserved e.g.: Which G-subformulae are eventually true? Salomon Sickert (TU München) 5 / 16
11 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Product of several automata Logical structure of the input formula is preserved e.g.: Which G-subformulae are eventually true? Smaller Systems 1 1 In most cases according to our experimental data; compared to the standard approach Salomon Sickert (TU München) 5 / 16
12 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Product of several automata Logical structure of the input formula is preserved e.g.: Which G-subformulae are eventually true? Smaller Systems 1 Bonus: Construction and correctness theorem verified in Isabelle/HOL 1 In most cases according to our experimental data; compared to the standard approach Salomon Sickert (TU München) 5 / 16
13 Deterministic ω-automata for LTL: A safraless, compositional, and mechanically verified construction Directly yields a deterministic system Product of several automata Logical structure of the input formula is preserved e.g.: Which G-subformulae are eventually true? Smaller Systems 1 Bonus: Construction and correctness theorem verified in Isabelle/HOL with code extraction 50% done 1 In most cases according to our experimental data; compared to the standard approach Salomon Sickert (TU München) 5 / 16
14 Experimental Data i {1,...,n} GFa i GFb i NBA DRA DTGRA LTL2BA ltl2dstar Rabinizer 3 n = 1 4 n = 2 14 n = 3 40 Salomon Sickert (TU München) 6 / 16
15 Experimental Data i {1,...,n} GFa i GFb i NBA DRA DTGRA LTL2BA ltl2dstar Rabinizer 3 n = n = 2 14 > 10 4 n = 3 40 > 10 6 Salomon Sickert (TU München) 6 / 16
16 Experimental Data i {1,...,n} GFa i GFb i NBA DRA DTGRA LTL2BA ltl2dstar Rabinizer 3 n = n = 2 14 > n = 3 40 > Salomon Sickert (TU München) 6 / 16
17 ω-words and LTL An ω-word is an infinite sequence: w = a 0 a 1 a 2 a Salomon Sickert (TU München) 7 / 16
18 ω-words and LTL An ω-word is an infinite sequence: w = a 0 a 1 a 2 a Definition (LTL Semantics, Negation-Normal-Form) :: α set word α ltl B w tt = True w ff = False w a = a w 0 w a = a w 0 w ϕ ψ = w ϕ w ψ w ϕ ψ = w ϕ w ψ Salomon Sickert (TU München) 7 / 16
19 ω-words and LTL An ω-word is an infinite sequence: w = a 0 a 1 a 2 a Definition (LTL Semantics, Negation-Normal-Form) :: α set word α ltl B w tt = True w ff = False w a = a w 0 w a = a w 0 w ϕ ψ = w ϕ w ψ w ϕ ψ = w ϕ w ψ w Fϕ = k. w k ϕ w Gϕ = k. w k ϕ w ψuϕ = k. w k ϕ j < k. w j ψ w Xϕ = w 1 ϕ Salomon Sickert (TU München) 7 / 16
20 ω-words and LTL An ω-word is an infinite sequence: w = a 0 a 1 a 2 a Definition (LTL Semantics, Negation-Normal-Form) :: α set word α ltl B w tt = True w ff = False w a = a w 0 w a = a w 0 w ϕ ψ = w ϕ w ψ w ϕ ψ = w ϕ w ψ w Fϕ = k. w k ϕ w Gϕ = k. w k ϕ w ψuϕ = k. w k ϕ j < k. w j ψ w Xϕ = w 1 ϕ Salomon Sickert (TU München) 7 / 16
21 ω-words and LTL An ω-word is an infinite sequence: w = a 0 a 1 a 2 a Definition (LTL Semantics, Negation-Normal-Form) :: α set word α ltl B w tt = True w ff = False w a = a w 0 w a = a w 0 w ϕ ψ = w ϕ w ψ w ϕ ψ = w ϕ w ψ w Fϕ = k. w k ϕ w Gϕ = k. w k ϕ w ψuϕ = k. w k ϕ j < k. w j ψ w Xϕ = w 1 ϕ Salomon Sickert (TU München) 7 / 16
22 Unfolding Modal Operators Fϕ XFϕ ϕ Gϕ XGϕ ϕ ψuϕ ϕ (ψ X(ψUϕ)) Salomon Sickert (TU München) 8 / 16
23 Co-Büchi Automata for G-free ϕ ϕ = a (b U c) Salomon Sickert (TU München) 9 / 16
24 Co-Büchi Automata for G-free ϕ ϕ ϕ = a (b U c) Salomon Sickert (TU München) 9 / 16
25 Co-Büchi Automata for G-free ϕ ϕ = a (b U c) ϕ a c (b X(bUc)) Salomon Sickert (TU München) 9 / 16
26 Co-Büchi Automata for G-free ϕ ϕ = a (b U c) ϕ a c (b X(bUc)) āb c buc Salomon Sickert (TU München) 9 / 16
27 Co-Büchi Automata for G-free ϕ ϕ = a (b U c) ϕ a c (b X(bUc)) āb c buc q 1 : a (b U c) āb c a + āc q 2 : b U c c b c b c ā b c true q 3 : tt q 4 : ff true Salomon Sickert (TU München) 9 / 16
28 Tackling the G-Operator Relaxed case: FGϕ w = FGϕ iff w i = ϕ for almost all i Reason: G-subformulae may be nested inside X, F, U. Salomon Sickert (TU München) 10 / 16
29 Automata for FGϕ where ϕ is G-free w =... a + āc āb c ā b c c q 2 b c b c q 3 q 4 Salomon Sickert (TU München) 11 / 16
30 Automata for FGϕ where ϕ is G-free w = abc... q 1 a + āc āb c q 2 c b c b c ā b c q 4 Salomon Sickert (TU München) 11 / 16
31 Automata for FGϕ where ϕ is G-free w = abc... q 1 a + āc āb c ā b c a + āc āb c ā b c c q 2 b c b c c q 2 b c b c q 4 q 3 q 4 Salomon Sickert (TU München) 11 / 16
32 Automata for FGϕ where ϕ is G-free w = abc āb c... q 1 q 1 a + āc āb c ā b c a + āc q 2 b c c b c āb c c b c b c ā b c a + āc āb c ā b c q 2 b c c b c q 4 q 3 q 4 q 3 q 4 Salomon Sickert (TU München) 11 / 16
33 Automata for FGϕ where ϕ is G-free w = abc āb c āb c... q 1 q 1 q 1 a + āc āb c ā b c a + āc q 2 b c c b c āb c c b c ā b c a + āc b c āb c c b c b c ā b c a + āc āb c ā b c q 2 b c c b c q 4 q 3 q 4 q 3 q 4 q 3 q 4 Salomon Sickert (TU München) 11 / 16
34 Automata for FGϕ where ϕ is G-free w = abc āb c āb c... q 1 q 1 q 1 a + āc āb c ā b c a + āc q 2 b c c b c āb c c b c ā b c a + āc b c āb c c b c b c ā b c a + āc āb c ā b c q 2 b c c b c q 4 q 3 q 4 q 3 q 4 q 3 q 4 3 a + āc 0 āb c 1, 2 c b c ā b c b c Salomon Sickert (TU München) 11 / 16
35 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ Salomon Sickert (TU München) 12 / 16
36 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ In every step a new token is placed in the initial state and all other tokens are moved according to the transition function. Salomon Sickert (TU München) 12 / 16
37 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ In every step a new token is placed in the initial state and all other tokens are moved according to the transition function. Deterministic Salomon Sickert (TU München) 12 / 16
38 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ In every step a new token is placed in the initial state and all other tokens are moved according to the transition function. Deterministic Accepts an ω-word w iff almost all tokens reach the final states Salomon Sickert (TU München) 12 / 16
39 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ In every step a new token is placed in the initial state and all other tokens are moved according to the transition function. Deterministic Accepts an ω-word w iff almost all tokens reach the final states Mojmir automata are blind to events that only happen finitely often Salomon Sickert (TU München) 12 / 16
40 Mojmir Automata 3 α 2 α 1 α 3 1, 2 β 2 β 3 β 1 Σ 0 Σ In every step a new token is placed in the initial state and all other tokens are moved according to the transition function. Deterministic Accepts an ω-word w iff almost all tokens reach the final states Mojmir automata are blind to events that only happen finitely often Salomon Sickert (TU München) 12 / 16
41 Going Further From Mojmir to Rabin Automata Salomon Sickert (TU München) 13 / 16
42 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Salomon Sickert (TU München) 13 / 16
43 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Abstraction with ranking functions for states and tokens Salomon Sickert (TU München) 13 / 16
44 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Abstraction with ranking functions for states and tokens Mojmir acceptance ( ) vs. Rabin acceptance (finite, )? Salomon Sickert (TU München) 13 / 16
45 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Abstraction with ranking functions for states and tokens Mojmir acceptance ( ) vs. Rabin acceptance (finite, )? Alternative definition for Mojmir acceptance Salomon Sickert (TU München) 13 / 16
46 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Abstraction with ranking functions for states and tokens Mojmir acceptance ( ) vs. Rabin acceptance (finite, )? Alternative definition for Mojmir acceptance Mojmir Automata for FGϕ for arbitrary ϕ Salomon Sickert (TU München) 13 / 16
47 Going Further From Mojmir to Rabin Automata Unbounded number of tokens? Abstraction with ranking functions for states and tokens Mojmir acceptance ( ) vs. Rabin acceptance (finite, )? Alternative definition for Mojmir acceptance Mojmir Automata for FGϕ for arbitrary ϕ Divide-and-conquer approach Construct for every G-subformula a separate automaton Instead of expanding G s rely on the other automata Intersection and Union of several Mojmir Automata Salomon Sickert (TU München) 13 / 16
48 Overview of the Construction Master-Transition-System LTL G-subformulae Product Generalised Rabin Mojmir Rabin Salomon Sickert (TU München) 14 / 16
49 Overview of the Construction Master-Transition-System LTL G-subformulae Product Generalised Rabin Mojmir Rabin The Master-Transition-System tracks a finite prefix of the ω-word. Salomon Sickert (TU München) 14 / 16
50 Overview of the Construction Master-Transition-System LTL G-subformulae Product Generalised Rabin Mojmir Rabin The Master-Transition-System tracks a finite prefix of the ω-word. Acceptance: 1 Guess the set of eventually true G-subformulae 2 Verify this guess using the Mojmir automata 3 Accept iff almost all the time this guess entails the current state of the master-transition-system Salomon Sickert (TU München) 14 / 16
51 Conclusion and Future Work The presented translation... preservers the logical structure of the formula is compositional Aggressive optimization can lead to huge space savings Some optimizations are already verified yields small deterministic ω-automata Salomon Sickert (TU München) 15 / 16
52 Conclusion and Future Work The presented translation... preservers the logical structure of the formula is compositional Aggressive optimization can lead to huge space savings Some optimizations are already verified yields small deterministic ω-automata Open Problems: Explore and formalize further optimizations Adapt construction to support: Alternation-free linear-time µ-calculus (contains LTL) Parity automata Salomon Sickert (TU München) 15 / 16
53 Getting More Information Javier Esparza, Jan Kretínský: From LTL to Deterministic Automata: A Safraless Compositional Approach. CAV 2014: pages From LTL to Deterministic Automata: A Safraless Compositional Approach Javier Esparza and Jan Křetínský p 2014 Institut für Informatik, Technische Universität München, Germany IST Austria Salomon Sickert (TU München) 16 / 16 Abstract. We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula ϕ. The automaton is
54 Getting More Information Javier Esparza, Jan Kretínský: From LTL to Deterministic Automata: A Safraless Compositional Approach. CAV 2014: pages Isabelle/HOL Formalisation To be submitted to the Archive of Formal Proofs - afp.sourceforge.net Available on request: sickert@in.tum.de From LTL to Deterministic Automata: A Safraless Compositional Approach Javier Esparza and Jan Křetínský p 2014 Institut für Informatik, Technische Universität München, Germany IST Austria Salomon Sickert (TU München) 16 / 16 Abstract. We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula ϕ. The automaton is
55 Getting More Information Javier Esparza, Jan Kretínský: From LTL to Deterministic Automata: A Safraless Compositional Approach. CAV 2014: pages Isabelle/HOL Formalisation To be submitted to the Archive of Formal Proofs - afp.sourceforge.net Available on request: sickert@in.tum.de Thank you for your attention! From LTL to Deterministic Automata: A Safraless Compositional Approach Javier Esparza and Jan Křetínský p 2014 Institut für Informatik, Technische Universität München, Germany IST Austria Salomon Sickert (TU München) 16 / 16 Abstract. We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula ϕ. The automaton is
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